Theorem cdleme39n 41302

Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 ∨ 𝑄 line. TODO: FIX COMMENT. 𝐸, 𝑌, 𝐺, 𝑍 serve as f(t), f(u), f_t(𝑅), f_t(𝑆). Put hypotheses of cdleme38n 41300 in convention of cdleme32sn1awN 41268. TODO see if this hypothesis conversion would be better if done earlier. (Contributed by NM, 15-Mar-2013.)

Hypotheses

Ref	Expression
cdleme39.l	⊢ ≤ = (le‘𝐾)
cdleme39.j	⊢ ∨ = (join‘𝐾)
cdleme39.m	⊢ ∧ = (meet‘𝐾)
cdleme39.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme39.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme39.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme39.e	⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme39.g	⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
cdleme39.y	⊢ 𝑌 = ((𝑢 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑢) ∧ 𝑊)))
cdleme39.z	⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑆 ∨ 𝑢) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme39n	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝐺 ≠ 𝑍)

Proof of Theorem cdleme39n

Step	Hyp	Ref	Expression
1		cdleme39.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		cdleme39.j	. . 3 ⊢ ∨ = (join‘𝐾)
3		cdleme39.m	. . 3 ⊢ ∧ = (meet‘𝐾)
4		cdleme39.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
5		cdleme39.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
6		cdleme39.u	. . 3 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
7		cdleme39.e	. . 3 ⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
8		cdleme39.y	. . 3 ⊢ 𝑌 = ((𝑢 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑢) ∧ 𝑊)))
9		eqid 2765	. . 3 ⊢ ((𝑡 ∨ 𝐸) ∧ 𝑊) = ((𝑡 ∨ 𝐸) ∧ 𝑊)
10		eqid 2765	. . 3 ⊢ ((𝑢 ∨ 𝑌) ∧ 𝑊) = ((𝑢 ∨ 𝑌) ∧ 𝑊)
11		eqid 2765	. . 3 ⊢ ((𝑅 ∨ ((𝑡 ∨ 𝐸) ∧ 𝑊)) ∧ (𝐸 ∨ ((𝑡 ∨ 𝑅) ∧ 𝑊))) = ((𝑅 ∨ ((𝑡 ∨ 𝐸) ∧ 𝑊)) ∧ (𝐸 ∨ ((𝑡 ∨ 𝑅) ∧ 𝑊)))
12		eqid 2765	. . 3 ⊢ ((𝑆 ∨ ((𝑢 ∨ 𝑌) ∧ 𝑊)) ∧ (𝑌 ∨ ((𝑢 ∨ 𝑆) ∧ 𝑊))) = ((𝑆 ∨ ((𝑢 ∨ 𝑌) ∧ 𝑊)) ∧ (𝑌 ∨ ((𝑢 ∨ 𝑆) ∧ 𝑊)))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdleme38n 41300	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ((𝑅 ∨ ((𝑡 ∨ 𝐸) ∧ 𝑊)) ∧ (𝐸 ∨ ((𝑡 ∨ 𝑅) ∧ 𝑊))) ≠ ((𝑆 ∨ ((𝑢 ∨ 𝑌) ∧ 𝑊)) ∧ (𝑌 ∨ ((𝑢 ∨ 𝑆) ∧ 𝑊))))
14		simp11 1222	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		simp12l 1305	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑃 ∈ 𝐴)
16		simp13l 1307	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑄 ∈ 𝐴)
17		simp22l 1311	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑅 ∈ 𝐴)
18		simp22r 1312	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑅 ≤ 𝑊)
19		simp311 1339	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
20		simp32l 1317	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))
21		cdleme39.g	. . . 4 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
22	1, 2, 3, 4, 5, 6, 7, 21, 9	cdleme39a 41301	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝐺 = ((𝑅 ∨ ((𝑡 ∨ 𝐸) ∧ 𝑊)) ∧ (𝐸 ∨ ((𝑡 ∨ 𝑅) ∧ 𝑊))))
23	14, 15, 16, 17, 18, 19, 20, 22	syl322anc 1425	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝐺 = ((𝑅 ∨ ((𝑡 ∨ 𝐸) ∧ 𝑊)) ∧ (𝐸 ∨ ((𝑡 ∨ 𝑅) ∧ 𝑊))))
24		simp23l 1313	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑆 ∈ 𝐴)
25		simp23r 1314	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑆 ≤ 𝑊)
26		simp312 1340	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑆 ≤ (𝑃 ∨ 𝑄))
27		simp33l 1319	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊))
28		cdleme39.z	. . . 4 ⊢ 𝑍 = ((𝑃 ∨ 𝑄) ∧ (𝑌 ∨ ((𝑆 ∨ 𝑢) ∧ 𝑊)))
29	1, 2, 3, 4, 5, 6, 8, 28, 10	cdleme39a 41301	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ (𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊))) → 𝑍 = ((𝑆 ∨ ((𝑢 ∨ 𝑌) ∧ 𝑊)) ∧ (𝑌 ∨ ((𝑢 ∨ 𝑆) ∧ 𝑊))))
30	14, 15, 16, 24, 25, 26, 27, 29	syl322anc 1425	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝑍 = ((𝑆 ∨ ((𝑢 ∨ 𝑌) ∧ 𝑊)) ∧ (𝑌 ∨ ((𝑢 ∨ 𝑆) ∧ 𝑊))))
31	13, 23, 30	3netr4d 3037	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ((𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≠ 𝑆) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) ∧ ((𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊) ∧ ¬ 𝑢 ≤ (𝑃 ∨ 𝑄)))) → 𝐺 ≠ 𝑍)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  lecple 17343  joincjn 18393  meetcmee 18394  Atomscatm 40099  HLchlt 40186  LHypclh 40820

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-proset 18376  df-poset 18395  df-plt 18410  df-lub 18426  df-glb 18427  df-join 18428  df-meet 18429  df-p0 18505  df-p1 18506  df-lat 18514  df-clat 18581  df-oposet 40012  df-ol 40014  df-oml 40015  df-covers 40102  df-ats 40103  df-atl 40134  df-cvlat 40158  df-hlat 40187  df-llines 40334  df-lplanes 40335  df-lvols 40336  df-lines 40337  df-psubsp 40339  df-pmap 40340  df-padd 40632  df-lhyp 40824

This theorem is used by:  cdleme40m  41303